A controversy, heated at times, ran on between Mendelians and biometricians until 1918. That year saw a seminal paper by RA Fisher in which he demonstrated that characters governed by a large number of independent Mendelian factors (polygenic characters) could display precisely the continuous nature, quantitative variation, and family correlations described by the biometricians. His paper can be read at https://digital.library. adelaide.edu.au/dspace/bitstream/2440/15097/1/9.pdf. It could not be described as easy reading. Later, DS Falconer extended this model to cover dichotomous non-Mendelian characters like birth defects. Fisher’s and Falconer’s analyses created a unified theoretical basis for human genetics. Here we set out their ideas, in a nonmathematical form. A more rigorous treatment can be found in textbooks of quantitative or population genetics.
Any variable quantitative character that depends on the additive action of a large number of small independent causes (whether genetic or not) will show a Normal (Gaussian) distribution in the population. Figure 1 gives a highly-simplified illustration of this for a genetic character.
Fig1. Successive approximations to a Gaussian distribution. The charts show the distribution in the population of a hypothetical continuous character that has a mean value of 100 units. The character is determined by the additive (co-dominant) effects of alleles. Each uppercase allele adds 5 units to the value, and each lowercase allele subtracts 5 units. All allele frequencies are 0.5. The character is determined by (A) a single locus, (B) two loci, and (C) three loci. (D) The addition of a minor amount of random (environmental or polygenic) variation produces a Gaussian curve.
In Figure 1 we suppose a character to depend on alleles at a single locus, then at two loci, then at three. As more loci are included, we see two consequences:
• The simple one-to-one relationship between genotype and phenotype disappears. Except for the extreme phenotypes, it is not possible to infer the genotype from the phenotype;
• As the number of loci increases, the distribution looks increasingly like a Gaussian curve. The addition of a little environmental variation would smooth out the three-locus distribution into a good Gaussian curve.
A more sophisticated treatment, allowing dominance and varying allele frequencies, leads to the same conclusions. Because relatives share genes, their phenotypes are correlated, and Fisher’s 1918 paper predicted the size of the correlation for different relationships.
Regression to the mean
A much-misunderstood feature, both of biometric data and of polygenic theory, is regression to the mean. Imagine, for the sake of example only, that IQ were a meaningful quantitative character in which all variation was entirely genetically determined. Figure 2 shows that in our simplified two-locus model, for each class of mothers, the average IQ of their children is halfway between the mother’s value and the population mean. This is regression to the mean—but its implications are often misinterpreted. Two common misconceptions are:
• After a few generations everybody will be exactly the same;
• If a character shows regression to the mean, it must be genetic.
• The overall distribution is the same in each generation;
• Regression works both ways: for each class of children, the average for their mothers is halfway between the children’s value and the population mean. This may sound paradoxical, but it can be confirmed by inspecting, for example, the right hand column of the bottom histogram in the figure (children of IQ 120). One quarter of their mothers have IQ 120, half 110, and one-quarter 100, making an average of 110.
Fig2. Regression to the mean. The same character as in Figure 5.20B: mean 100, determined by co-dominant alleles A, a, B, and b at two loci, all allele frequencies = 0.5. Top: distribution in a series of mothers. Middle: distributions in children of each class of mothers, assuming random mating. Bottom: summed distribution in the children. Note that: (1) the distribution in the children is the same as the distribution in the mothers; (2) for each class of mothers, the mean for their children is halfway between the mothers’ value and the population mean (100); and (3) for each class of children (bottom), the mean for their mothers is halfway between the children’s value and the population mean.
Regarding the second of these beliefs, regression to the mean is not a genetic mechanism but a purely statistical phenomenon. Whether the determinants of IQ are genetic, environmental, or any mix of the two, if we take an exceptional group of mothers (for example, those with an IQ of 120), then these mothers must have had an exceptional set of determinants. If we take a second group who share half those determinants (their children, their sibs, or either of their parents), the average phenotype in this second group will deviate from the population mean by half as much. Genetics provides the figure of one-half—it is because each child inherits one-half of his or her genes from their mother that the average IQ of the children (in this simple model) is halfway between the mother’s IQ and the population mean—but genetics does not supply the principle of regression.
The simplified model has hidden assumptions
In the simple model of Figure 2 there is a hidden assumption: that there is random mating. For each class of mothers, the average IQ of their husbands is assumed to be 100. Thus, the average IQ of the children is actually the mid-parental IQ, as common sense would suggest. In the real world, highly-intelligent women tend to marry men of above average intelligence (assortative mating). The regression would therefore be less than halfway to the population mean, if IQ were a purely genetic character—which, of course, it isn’t.
A second assumption of our simplified model is that there is no dominance. Each per son’s phenotype is assumed to be the simple sum of the contribution of each allele at each relevant locus. If we allow dominance, the effect of some of a parent’s genes will be masked by dominant alleles and invisible in their phenotype, but they can still be passed on and can affect the child’s phenotype. Given dominance, the expectation for the child is no longer the mid-parental value. Our best guess about the likely phenotypic effect of the masked recessive alleles is obtained by looking at the rest of the population. Therefore, the child’s expected phenotype will be displaced from the mid-parental value toward the population mean. How far it will be displaced depends on how important dominance is in determining the phenotype.
